3.163 \(\int \cot ^3(a+b x) \csc (a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]

[Out]

csc(b*x+a)/b-1/3*csc(b*x+a)^3/b

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2606} \[ \frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*x]^3*Csc[a + b*x],x]

[Out]

Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

\begin {align*} \int \cot ^3(a+b x) \csc (a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (-1+x^2\right ) \, dx,x,\csc (a+b x)\right )}{b}\\ &=\frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 1.00 \[ \frac {\csc (a+b x)}{b}-\frac {\csc ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*x]^3*Csc[a + b*x],x]

[Out]

Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)

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fricas [A]  time = 0.42, size = 38, normalized size = 1.46 \[ \frac {3 \, \cos \left (b x + a\right )^{2} - 2}{3 \, {\left (b \cos \left (b x + a\right )^{2} - b\right )} \sin \left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^4,x, algorithm="fricas")

[Out]

1/3*(3*cos(b*x + a)^2 - 2)/((b*cos(b*x + a)^2 - b)*sin(b*x + a))

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giac [A]  time = 0.18, size = 25, normalized size = 0.96 \[ \frac {3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^4,x, algorithm="giac")

[Out]

1/3*(3*sin(b*x + a)^2 - 1)/(b*sin(b*x + a)^3)

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maple [B]  time = 0.02, size = 60, normalized size = 2.31 \[ \frac {-\frac {\cos ^{4}\left (b x +a \right )}{3 \sin \left (b x +a \right )^{3}}+\frac {\cos ^{4}\left (b x +a \right )}{3 \sin \left (b x +a \right )}+\frac {\left (2+\cos ^{2}\left (b x +a \right )\right ) \sin \left (b x +a \right )}{3}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^3/sin(b*x+a)^4,x)

[Out]

1/b*(-1/3*cos(b*x+a)^4/sin(b*x+a)^3+1/3*cos(b*x+a)^4/sin(b*x+a)+1/3*(2+cos(b*x+a)^2)*sin(b*x+a))

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maxima [A]  time = 0.30, size = 25, normalized size = 0.96 \[ \frac {3 \, \sin \left (b x + a\right )^{2} - 1}{3 \, b \sin \left (b x + a\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^3/sin(b*x+a)^4,x, algorithm="maxima")

[Out]

1/3*(3*sin(b*x + a)^2 - 1)/(b*sin(b*x + a)^3)

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mupad [B]  time = 0.43, size = 22, normalized size = 0.85 \[ \frac {{\sin \left (a+b\,x\right )}^2-\frac {1}{3}}{b\,{\sin \left (a+b\,x\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^3/sin(a + b*x)^4,x)

[Out]

(sin(a + b*x)^2 - 1/3)/(b*sin(a + b*x)^3)

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sympy [A]  time = 2.26, size = 42, normalized size = 1.62 \[ \begin {cases} \frac {2}{3 b \sin {\left (a + b x \right )}} - \frac {\cos ^{2}{\left (a + b x \right )}}{3 b \sin ^{3}{\left (a + b x \right )}} & \text {for}\: b \neq 0 \\\frac {x \cos ^{3}{\relax (a )}}{\sin ^{4}{\relax (a )}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**3/sin(b*x+a)**4,x)

[Out]

Piecewise((2/(3*b*sin(a + b*x)) - cos(a + b*x)**2/(3*b*sin(a + b*x)**3), Ne(b, 0)), (x*cos(a)**3/sin(a)**4, Tr
ue))

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